Place peaches in a blender or food processor; process until smooth. Press peach puree through a fine sieve into a bowl; discard solids.
Combine lime juice, sugar, and mint in a large measuring cup; crush juice mixture with a wooden muddler or the back of a long spoon.
Set aside for about 20 minutes (this give the sugar time to dissolve). Stir in club soda and peach juice. Serve over crushed ice. Garnish with mint sprigs, if desired.